- a factor is an integer that divides another integer with no remainder leftover
To illustrate the idea of a factor, consider the numbers 15. Here, 5 is a factor of 15 because $15 \div 5 = 3$ has a zero remainder.
- common divisor
- a common divisor is a factor of two or more integers
3 is a common divisor of 15 and 30 since it divides both numbers with no remainder. -3 is also a factor of 15 and 30.
- greatest common divisor
- the greatest common divisor is the largest factor of two integers
3 is not the greatest common divisor of 15 and 30 since there is a larger factor that divides both numbers, 15. The number 15 divides 15 and 30 without any remainder.
- an integer, $n$, is a multiple of a given integer if $n$ divided by the given integer has a zero remainder. If so, we say $n$ is a multiple of the given integer.
For example, 15 is a multiple of 5 because 15 $\div$ 5 has a zero remainder. 5 is not a multiple of 15 because 5 $\div$ 15 does not have a zero remainder. However, -30 is a multiple of 15 because -30 $\div$ 15 has a zero remainder (count backwards).
- common multiple
- an integer, $n$, is a common multiple of two given integers, if the two given integers divide $n$ with zero remainder.
- least common multiple
- a positive integer, $n$, is the least common multiple of two given integers if $n$ is the smallest common multiple of two given integers.
For example, 30 is a common multiple of 5 and 15 since 30 $\div$ 5 and 30 $\div$ 15 have zero remainders. However, 30 is not the least common multiple because of 15. Here, 15 $\div$ 15 and 15 $\div$ 5 have zero remainders and 15 $\lt$ 30. Please note, -15 is not the least common multiple of 5 and 15 because a least common multiple must be positive.